The role of feedback circuits: Positive feedback circuits are a necessary condition for positive real eigenvalues of the Jacobian matrix

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Abstract
This talk deals with the biological roles of feedback loops and their combinations. It has progressively become clear that a negative loop is a necessary (not sufficient) condition for a stable periodic behaviour and that a positive loop is a necessary (not sufficient) condition for multistationarity. Of these old conjectures (e.g. Stud. phys. theor. Chem. 28, 307 (1983)), the second has now two formal demonstrations by others (Plahte et al., Snoussi). Here, we remark that this statement can be generalised as follows. The presence of a positive loop (in the Jacobian matrix of the system) is a necessary condition for having at least one real positive root in the characteristic equation (this can be shown to directly derive from the Descartes rule) and at least one real positive root in turn is a necessary condition for multistationarity.